Synthetic Division Calculator

The calculator will divide the polynomial by the binomial using synthetic division, with steps shown.

Show Instructions
  • In general, you can skip the multiplication sign, so `5x` is equivalent to `5*x`.
  • In general, you can skip parentheses, but be very careful: e^3x is `e^3x`, and e^(3x) is `e^(3x)`.
  • Also, be careful when you write fractions: 1/x^2 ln(x) is `1/x^2 ln(x)`, and 1/(x^2 ln(x)) is `1/(x^2 ln(x))`.
  • If you skip parentheses or a multiplication sign, type at least a whitespace, i.e. write sin x (or even better sin(x)) instead of sinx.
  • Sometimes I see expressions like tan^2xsec^3x: this will be parsed as `tan^(2*3)(x sec(x))`. To get `tan^2(x)sec^3(x)`, use parentheses: tan^2(x)sec^3(x).
  • Similarly, tanxsec^3x will be parsed as `tan(xsec^3(x))`. To get `tan(x)sec^3(x)`, use parentheses: tan(x)sec^3(x).
  • From the table below, you can notice that sech is not supported, but you can still enter it using the identity `sech(x)=1/cosh(x)`.
  • If you get an error, double check your expression, add parentheses and multiplication signs where needed, and consult the table below.
  • All suggestions and improvements are welcome. Please leave them in comments.
The following table contains the supported operations and functions:
TypeGet
Constants
ee
pi`pi`
ii (imaginary unit)
Operations
a+ba+b
a-ba-b
a*b`a*b`
a^b, a**b`a^b`
sqrt(x), x^(1/2)`sqrt(x)`
cbrt(x), x^(1/3)`root(3)(x)`
root(x,n), x^(1/n)`root(n)(x)`
x^(a/b)`x^(a/b)`
abs(x)`|x|`
Functions
e^x`e^x`
ln(x), log(x)ln(x)
ln(x)/ln(a)`log_a(x)`
Trigonometric Functions
sin(x)sin(x)
cos(x)cos(x)
tan(x)tan(x), tg(x)
cot(x)cot(x), ctg(x)
sec(x)sec(x)
csc(x)csc(x), cosec(x)
Inverse Trigonometric Functions
asin(x), arcsin(x), sin^-1(x)asin(x)
acos(x), arccos(x), cos^-1(x)acos(x)
atan(x), arctan(x), tan^-1(x)atan(x)
acot(x), arccot(x), cot^-1(x)acot(x)
asec(x), arcsec(x), sec^-1(x)asec(x)
acsc(x), arccsc(x), csc^-1(x)acsc(x)
Hyperbolic Functions
sinh(x)sinh(x)
cosh(x)cosh(x)
tanh(x)tanh(x)
coth(x)coth(x)
1/cosh(x)sech(x)
1/sinh(x)csch(x)
Inverse Hyperbolic Functions
asinh(x), arcsinh(x), sinh^-1(x)asinh(x)
acosh(x), arccosh(x), cosh^-1(x)acosh(x)
atanh(x), arctanh(x), tanh^-1(x)atanh(x)
acoth(x), arccoth(x), cot^-1(x)acoth(x)
acosh(1/x)asech(x)
asinh(1/x)acsch(x)

Divide (dividend):

By (divisor):

Binomial (of the form `ax+b`)

Write all suggestions in comments below.

Solution

Your input: find $$$\frac{x^{4} - 5 x^{3} + 7 x^{2} - 34 x - 1}{x - 5}$$$ using synthetic division.

Write the problem in a division-like format.

To do this:

  • Take the constant term of the divisor with the opposite sign and write it to the left.
  • Write the coefficients of the dividend to the right.

$$$\begin{array}{c|ccccc}&x^{4}&x^{3}&x^{2}&x^{1}&x^{0}\\5&1&-5&7&-34&-1\\&&\\\hline&\end{array}$$$

Step 1

Write down the first coefficient without changes:

$$$\begin{array}{c|rrrrr}5&\color{Chocolate }{1}&-5&7&-34&-1\\&&\\\hline&\color{Chocolate }{1}\end{array}$$$

Step 2

Multiply the entry in the left part of the table by the last entry in the result row (under the horizontal line).

Add the obtained result to the next coefficient of the dividend, and write down the sum.

$$$\begin{array}{c|rrrrr}\color{Magenta}{5}&1&\color{SaddleBrown}{-5}&7&-34&-1\\&&\color{Magenta}{5} \cdot \color{Chocolate }{1}=\color{Red}{5}\\\hline&\color{Chocolate }{1}&\left(\color{SaddleBrown}{-5}\right)+\color{Red}{5}=\color{Green}{0}\end{array}$$$

Step 3

Multiply the entry in the left part of the table by the last entry in the result row (under the horizontal line).

Add the obtained result to the next coefficient of the dividend, and write down the sum.

$$$\begin{array}{c|rrrrr}\color{Magenta}{5}&1&-5&\color{GoldenRod}{7}&-34&-1\\&&5&\color{Magenta}{5} \cdot \color{SaddleBrown}{0}=\color{Red}{0}\\\hline&1&\color{SaddleBrown}{0}&\color{GoldenRod}{7}+\color{Red}{0}=\color{Green}{7}\end{array}$$$

Step 4

Multiply the entry in the left part of the table by the last entry in the result row (under the horizontal line).

Add the obtained result to the next coefficient of the dividend, and write down the sum.

$$$\begin{array}{c|rrrrr}\color{Magenta}{5}&1&-5&7&\color{BlueViolet}{-34}&-1\\&&5&\color{Magenta}{5} \cdot \color{GoldenRod}{7}=\color{Red}{35}\\\hline&1&0&\color{GoldenRod}{7}&\left(\color{BlueViolet}{-34}\right)+\color{Red}{35}=\color{Green}{1}\end{array}$$$

Step 5

Multiply the entry in the left part of the table by the last entry in the result row (under the horizontal line).

Add the obtained result to the next coefficient of the dividend, and write down the sum.

$$$\begin{array}{c|rrrrr}\color{Magenta}{5}&1&-5&7&-34&\color{Brown}{-1}\\&&5&35&\color{Magenta}{5} \cdot \color{BlueViolet}{1}=\color{Red}{5}\\\hline&1&0&7&\color{BlueViolet}{1}&\left(\color{Brown}{-1}\right)+\color{Red}{5}=\color{Green}{4}\end{array}$$$

We've completed the table and obtained the following resulting coefficients: $$$1, 0, 7, 1, 4$$$.

All the coefficients except the last one are the coefficients of the quotient, the last coefficient is the remainder.

Thus, the quotient is $$$x^{3}+0 x^{2}+7 x+1$$$, and the remainder is $$$4$$$.

Therefore, $$$\frac{x^{4} - 5 x^{3} + 7 x^{2} - 34 x - 1}{x - 5}=x^{3} + 7 x + 1+\frac{4}{x - 5}$$$.

Answer: $$$\frac{x^{4} - 5 x^{3} + 7 x^{2} - 34 x - 1}{x - 5}=x^{3} + 7 x + 1+\frac{4}{x - 5}$$$.